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Risk and Return: Rates of Return and Measuring Variance

This chapter discusses rates of return, including single-period returns and dollar-weighted returns. It also covers measuring mean and variance of returns, as well as the characteristics of probability distributions. Additionally, the chapter explores real vs. nominal rates, allocating capital between risky and risk-free assets, and using leverage with the capital allocation line.

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Risk and Return: Rates of Return and Measuring Variance

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  1. Chapter5 Risk and Return: Past and Prologue

  2. Rates of Return: Single Period HPR = Holding Period Return P1 = Ending price P0 = Beginning price D1 = Dividend during period one

  3. Rates of Return: Single Period Example Ending Price = 24 Beginning Price = 20 Dividend = 1 HPR = ( 24 - 20 + 1 )/ ( 20) = 25%

  4. Data from Text Example p. 154 1 2 3 4 Assets(Beg.) 1.0 1.2 2.0 .8 HPR .10 .25 (.20) .25 TA (Before Net Flows 1.1 1.5 1.6 1.0 Net Flows 0.1 0.5 (0.8) 0.0 End Assets 1.2 2.0 .8 1.0

  5. Returns Using Arithmetic and Geometric Averaging Arithmetic ra = (r1 + r2 + r3 + ... rn) / n ra = (.10 + .25 - .20 + .25) / 4 = .10 or 10% Geometric rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1 rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1 = (1.5150) 1/4 -1 = .0829 = 8.29%

  6. Dollar Weighted Returns Internal Rate of Return (IRR) - the discount rate that results present value of the future cash flows being equal to the investment amount • Considers changes in investment • Initial Investment is an outflow • Ending value is considered as an inflow • Additional investment is a negative flow • Reduced investment is a positive flow

  7. Dollar Weighted Average Using Text Example Net CFs 1 2 3 4 $ (mil) - .1 - .5 .8 1.0 Solving for IRR 1.0 = -.1/(1+r)1 + -.5/(1+r)2 + .8/(1+r)3 + 1.0/(1+r)4 r = .0417 or 4.17%

  8. Quoting Conventions APR = annual percentage rate (periods in year) X (rate for period) EAR = effective annual rate ( 1+ rate for period)Periods per yr - 1 Example: monthly return of 1% APR = 1% X 12 = 12% EAR = (1.01)12 - 1 = 12.68%

  9. Characteristics of Probability Distributions 1) Mean: most likely value 2) Variance or standard deviation 3) Skewness * If a distribution is approximately normal, the distribution is described by characteristics 1 and 2

  10. Normal Distribution s.d. s.d. r Symmetric distribution

  11. Skewed Distribution: Large Negative Returns Possible Median Negative Positive r

  12. Skewed Distribution: Large Positive Returns Possible Median Negative r Positive

  13. S E ( r ) = p ( s ) r ( s ) s Measuring Mean: Scenario or Subjective Returns Subjective returns p(s) = probability of a state r(s) = return if a state occurs 1 to s states

  14. Numerical Example: Subjective or Scenario Distributions StateProb. of State rin State 1 .1 -.05 2 .2 .05 3 .4 .15 4 .2 .25 5 .1 .35 E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35) E(r) = .15

  15. S 2 Variance = p ( s ) [ r - E ( r )] s s Measuring Variance or Dispersion of Returns Subjective or Scenario Standard deviation = [variance]1/2 Using Our Example: Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2] Var= .01199 S.D.= [ .01199] 1/2 = .1095

  16. Real vs. Nominal Rates Fisher effect: Approximation nominal rate = real rate + inflation premium R = r + i or r = R - i Example r = 3%, i = 6% R = 9% = 3% + 6% or 3% = 9% - 6% Fisher effect: Exact r = (R - i) / (1 + i) 2.83% = (9%-6%) / (1.06)

  17. Annual Holding Period ReturnsFrom Figure 6.1 of Text Geom Arith Stan. Series Mean% Mean% Dev.% Lg Stk 11.01 13.00 20.33 Sm Stk 12.46 18.77 39.95 LT Gov 5.26 5.54 7.99 T-Bills 3.75 3.80 3.31 Inflation 3.08 3.18 4.49

  18. Annual Holding Period Risk Premiums and Real Returns Risk Real Series Premiums% Returns% Lg Stk 9.2 9.82 Sm Stk 14.97 15.59 LT Gov 1.74 2.36 T-Bills --- 0.62 Inflation --- ---

  19. Allocating Capital Between Risky & Risk-Free Assets • Possible to split investment funds between safe and risky assets • Risk free asset: proxy; T-bills • Risky asset: stock (or a portfolio)

  20. Allocating Capital Between Risky & Risk-Free Assets (cont.) • Issues • Examine risk/ return tradeoff • Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets

  21. rf = 7% srf = 0% E(rp) = 15% sp = 22% y = % in p (1-y) = % in rf Example Using the Numbers in Chapter 6 (pp 171-173)

  22. E(rc) = yE(rp) + (1 - y)rf rc = complete or combined portfolio For example, y = .75 E(rc) = .75(.15) + .25(.07) = .13 or 13% Expected Returns for Combinations

  23. Possible Combinations E(r) E(rp) = 15% P rf = 7% F 0 s 22%

  24. s Since = 0, then rf = y c p Variance on the Possible Combined Portfolios s s

  25. If y = .75, then = .75(.22) = .165 or 16.5% c If y = 1 = 1(.22) = .22 or 22% c If y = 0 = 0(.22) = .00 or 0% c Combinations Without Leverage s s s

  26. Using Leverage with Capital Allocation Line Borrow at the Risk-Free Rate and invest in stock (while not really possible, lets assume we can do it) Using 50% Leverage rc = (-.5) (.07) + (1.5) (.15) = .19 sc = (1.5) (.22) = .33 Note that we assume the T-bill is totally risk free (bear with me again)

  27. Capital Allocation Line CAL: (Capital Allocation Line) E(r) This graph is the risk return combination available by choosing different values of y. Note we have E(r) and variance on the axis. P E(rp) = 15% Risk premium E(rp) - rf = 8% ) S = 8/22 rf = 7% F Slope: Reward to variability ratio: ratio of risk premium to std. dev. s 0 P = 22%

  28. Problem 9: Portfolio Return Stock price and dividend history Year Beginning stock price Dividend Yield 2001 $100 $4 2002 110 $4 2003 90 $4 2004 95 $4 An investor buys three shares at the beginning of 2001, buys another 2 at the beginning of 2002, sells 1 share at the beginning of 2003, and sells all 4 remaining at the beginning of 2004. A. What are the arithmetic and geometric average time-weighted rates of return? B. What is the dollar weighted rate of return?

  29. Answer • Time weighted return • 2001 (110-100+4)/100 = 14% • 2002 (90-110+4)/110 = - 14.6% • 2003 (95-90+4)/90 = 10% • Arithmetic mean return (14-14.6+10)/3 = 3.13% • Geometric mean return (1+.14)*(1-.146)*(1+.1)]1/3 = 1.078.33 –1 = 2.3%

  30. Problem 11: Risk Premiums • Using the historical risk premiums as your guide from the chart earlier, what is your estimate of the expected annual HPR on the S&P500 stock portfolio if the current risk-free interest rate is 5.0%. What does the risk premium represent?

  31. Answer For the period of 1926- 2004 the large cap stocks returned 10.0%, less t-bills of 3.7% gives a risk premium of 6.3%. • If the current risk free rate is 5.0%, then • E(r) = Risk free rate + risk premium • E(r) = 5.0% + 6.3% = 11.3% • The risk premium represents the additional return that is required to compensate you for the additional risk you are taking on to invest in this asset class.

  32. Problem 12: Client Portfolios • You manage a risky portfolio with an expected return of 12% and a standard deviation of 25%. The T-bill rate is 4%. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. What is the expected return and standard deviation of your client’s portfolio? • Clients Fund E(r) (expected return) =.7 x 12% + .3 x 4% = 9.6% σ (standard deviation) = .7 x .25 = 17.5%

  33. Problem 13: Portfolio Allocations • Suppose your risky portfolio includes investments in the following proportions. What are the investment proportions in your clients portfolio Stock A 27% Stock B 33% Stock C 40% • Investment proportions: T-bills = 30% Stock A = .7 x 27% = 18.9% Stock B = .7 x 33% = 23.1% Stock A = .7 x 40% = 28.0% Check: 30 + 18.9 + 23.1 + 28 = 100%

  34. Problem 14: Reward to Variability C. What is the reward-to-variability ratio (s) of your risky portfolio and your clients portfolio? • Reward to Variability (risk premium / standard deviation) • Fund = (12.0% – 4%) / 25 = .32 • Client = (9.6% – 4%) / 17.5 = .32

  35. Problem 15: The CAL Line D. Draw the CAL of your portfolio. What is the slope of the CAL? Slope of the CAL line % Slope = .3704 17 P 14 Client Standard Deviation 18.9 27 7

  36. Problem 16: Maximizing Standard Deviation Suppose the client in Problem 12 prefers to invest in your portfolio a proportion (y) that maximizes the expected return on the overall portfolio subject to the constraint that the overall portfolio’s standard deviation will not exceed 20%. What is the investment proportion? What is the expected return on the portfolio?

  37. Answer Portfolio standard deviation 20% = (y) x 25% Y = 20/25 = 80.0% Mean return = (.80 x 12%) + (.20 x 4%) = 10.4%

  38. Problem 17: Increasing Stock Volatility • What do you think would happen to the expected return on stocks if investors perceived an increased volatility of stocks due to some recent event, i.e. Hurricane Katrina?

  39. Answer • Assuming no change in risk aversion, investors perceiving higher risk will demand a higher risk premium to hold the same portfolio they held before. If we assume the risk-free rate is unchanged, the increase in the risk premium would require a higher expected rate of return in the equity market.

  40. A. Do you understand rates of return? B. Do you know how to calculate return using scenario, probabilities, and other key statistics used to describe your portfolio? C. Do you understand the implications of using a risky and a risk-free asset in a portfolio? Review of Objectives

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